Some remarks on Bell-state measurement, its uses, and on quantum error correction Measurement as action: gloss of KLM Related pre- and postselected effects: quantum-interferometric effective nonlinearities importance of both preparation & projection difference between detection & measurement Summary: Measurement is more general than projection One can measure postselected "subensembles" Measurement has important physical effects 16 Dec 2003 (and Happy Holidays!) Some final words about quantum measurement...

Information and measurement Any measurement on a qubit (two-level system) yields at most 1 bit of info. On the other hand, a full specification of the state (density matrix) of a qubit involves 3 independent real parameters (coordinates on Bloch/Poincaré sphere); this is in principle an infinite amount of information. How much information can be stored or transferred using qubits? Measure & reproduce – only one classical bit results from the measurement, and this is all which can be reproduced. "No cloning": cannot make faithful copies of unknown, non-orthogonal quantum states, because { |b>} = { } 2 and unitary evolution preserves the inner product. [Wooters & Zurek, Nature 299, 802 (1982).] (N.B.: Applies to unitary evolution. With projection, one can for instance distinguish 0 from 45 sometimes, and then reproduce the exact state – but notice, still only one classical bit's worth of information.)

Dense coding & Teleportation Observation: a pair of entangled photons has four orthogonal basis states – the Bell states – but they can be connected by operations on a single photon. Thus sending that single photon to a partner who already possesses the other entangled photon allows one to convey 2 classical bits using a single photon. Bennett & Wiesner, PRL 69, 2881 (1992) The Bell state basis: flip phase flip pol. single-photon operations:

Log 2 (3) bits in a single photon To extract both bits, one would need to distinguish all 4 Bell states – this can't be done with linear optics, but 3 of the 4 can. (Recall: a Hong-Ou-Mandel already filters out the singlet) Mattle et al., PRL 76, 4656 (1996)

Quantum Teleportation (And the other three results just leave Bob with a unitary operation to do) Bennett et al., Phys. Rev. Lett. 70, 1895 (1993) BSM If BSM finds A & S in a singlet state, then we know they have opposite polarisation. Let Bob know the result. If S and I were opposite, and A and S were opposite, then I = A! singlet states S and I have opposite polarisations S I Alice Bob A (unknown state)

Quantum Teleportation (expt) Bouwmeester et al., Nature 390, 575 (1997)

One striking aspect of teleportation Alice's photon and Bob's have no initial relationship – Bob's could be in any of an infinite positions on the Poincaré sphere. The Bell-state measurement collapses photon S (and hence Bob's photon I) into one of four particular states – states with well-defined relationships to Alice's initial photon. Thus this measurement transforms a continuous, infinite range of possibilities (which we couldn't detect, let alone communicate to Bob) into a small discrete set. All possible states can be teleported, by projecting the continuum onto this complete set.

Quantum Error Correction In classical computers, small errors are continuously corrected – built-in dissipation pulls everything back towards a "1" or a "0". Recall that quantum computers must avoid dissipation and irreversibility. How, then, can errors be avoided? A bit could be anywhere on the Poincaré sphere – and an error could in principle move it anywhere else. Can we use measurement to reduce the error symptoms to a discrete set, à la teleportation? Yes: if you measure whether or not a bit flipped, you get either a "YES" or a "NO", and can correct it in the case of "YES". As in dense coding, the phase degree of freedom is also important, but you can similary measure whether or not the phase was flipped, and then correct that. Any possible error can be collapsed onto a "YES" or "NO" for each of these.

The four linearly independent errors

Q. error correction: Shor's 3-bit code Encode: a|0> + b|1>  a|000> + b|111> Symptom State i1i2i1i2 i1 i3i1 i3 Nothing happens a|000> + b|111> 00 i 1 flipsa|100> + b|011> 11 i 2 flipsa|010> + b|101> 10 i 3 flipsa|001> + b|110> 01 In case of bit flips, use redundancy – it's unlikely that more than 1 bit will flip at once, so we can use "majority rule"... BUT: we must not actually measure the value of the bits! And now just flip i 1 back if you found that it was flipped – note that when you measure which of these four error syndromes occurred, you exhaust all the information in the two extra bits, and no record is left of the value of i 1 ! errors i1i1 i2i2 i3i3 i 1  i 2 i 1  i 3

special |  i > a|0> + b|1> + c|2>a|0> + b|1> – c|2> The dream of optical quantum computing INPUT STATE ANCILLA TRIGGER (postselection) OUTPUT STATE particular |  f > MAGIC MIRROR: Acts differently if there are 2 photons or only 1. In other words, can be a “transistor,” or “switch,” or “quantum logic gate”... But real nonlinear interactions are typically times too weak to do this! What can one do with purely "linear" optics?

Hong-Ou-Mandel as interaction? |H> a|H>+b|V> If I detect a "trigger" photon here......then anything which comes out here must have the opposite polarisation. Two non-interacting photons became entangled, not only by meeting at a beam-splitter, but by being found on opposite sides (postselection). Choosing the state of one can determine which states of the other are allowed to be reflected (if we only pay attention to cases where coincidences occur.)

|1> a|0> + b|1> + c|2>a'|0> + b'|1> + c'|2> The germ of the KLM idea INPUT STATE ANCILLA TRIGGER (postselection) OUTPUT STATE |1> In particular: with a similar but somewhat more complicated setup, one can engineer a |0> + b |1> + c |2>  a |0> + b |1> – c |2> ; effectively a huge self-phase modulation (  per photon). More surprisingly, one can efficiently use this for scalable QC. KLM Nature 409, 46, (2001); Cf. Kaoru's experiment; Sara's experiment; experiments by Franson et al., White et al.,...

(What you really have to do)

The mad, mad idea of Jim Franson Nonlinear coefficients scale linearly with the number of atoms. Could the different atoms' effects be made to add coherently, providing an N 2 enhancement (where N might be )? atom 1 atom 2 11 11 22 22 Appears to violate local energy conservation... but consists of perfectly reasonable Feynman diagrams, with energy conserved in final state. {Controversy regarding some magic cancellations....} Each of N(N-1)/2 pairs of atoms should contribute. Franson proposes that this can lead to immense nonlinearities. No conclusive data. J.D. Franson, Phys. Rev. Lett 78, 3852 (1997)

John Sipe's suggestion Franson's proposal to harness photon-exchange terms investigates the effect on the real index of refraction (virtual intermediate state). Why not first search for such effects on real intermediate states (absorption)? Conclusion: exchange effects do matter: Probability of two-photon absorption may be larger than product of single-photon abs. prob's. Caveat: the effect indeed goes as N 2,... but N is the photon number (2) and not the atom number (10 13 ) ! Two-photon absorption (by these single-photon absorbers) is inter- ferometrically enhanced if the photons begin distinguishable, but are indistinguishable to the absorber: T 2 >  >  c

Ugly data,but it works. Roughly a 4% drop observed in 2-photon transmission when the photons are delayed relative to one another. Complicated by other effects due to straightforward frequency correlations between photons (cf. Wong, Sergienko, Walmsley,...), as well as correlations between spatial and spectral mode. Resch et al. quant-ph/

What was the setup? Type-II SPDC + birefringent delay + 45 o polarizer produces delayed pairs. Use a reflective notch filter as absorbing medium, and detect remaining pairs. This is just a Hong-Ou-Mandel interferometer, with detection in a complementary mode. Although the filter is placed after the output, this is irrelevant for a linear system. Interpretations: Our "suppressed" two-photon reflection is merely the ratio of two different interference patterns; the modified spectrum broadens the pattern. Yet photons which reach the filter in pairs really do not behave independently. The HOM interference pattern is itself a manifestation of photon exchange effects.

Another approach to 2-photon interactions... Ask: Is SPDC really the time-reverse of SHG? The probability of 2 photons upconverting in a typical nonlinear crystal is roughly 10  (as is the probability of 1 photon spontaneously down-converting). (And if so, then why doesn't it exist in classical e&m?)

Quantum Interference

2-photon "Switch": experiment

(57% visibility) Suppression/Enhancement of Spontaneous Down-Conversion

Photon-photon transmission switch On average, less than one photon per pulse. One photon present in a given pulse is sufficient to switch off transmission. The photons upconvert with near-unit eff. (Peak power approx. mW/cm 2 ). The blue pump serves as a catalyst, enhancing the interaction by

Switchiness ("Nonlinearity")

Controlled-phase switch Resch et al, Phys. Rev. Lett. 89, (2002)

Fringe data with and w/o postsel.

...but it actually is true

So: problem solved? This switch relies on interference; the input state must have a specific phase. Single photons don't have well-defined phase; the switch does not work on Fock states. The phase shifts if and only if a control photon is present-- so long as we make sure not to know in advance whether or not it is present. Preparation: weak coherent state with definite phase Post-selection: a "trigger" photon present in that state Result: a strong nonlinearity mediated by that state, between preparation and postselection. Not entirely...

We have shown theoretically that a polarisation version could be used for Bell-state determination (and, e.g., dense coding)… a task known to be impossible with LO. [Resch et al., quant-ph/ ] Like the case of non-orthogonal state discrimination, however, this is not the same as projective measurement. If you promise me to give me one of the Bell states, I can tell you which one I received; but I can't span the space of all possible states by projecting onto one of the four. (No teleportation.) Detection is not all there is to measurement...

SUMMARY (Cartoon stolen from Jonathan Dowling) Measurement is more general than projection One can measure postselected "subensembles" Measurement has important physical effects